Homology, Homotopy and Applications

Volume 20 (2018)

Number 1

Tensoring with the Frobenius endomorphism

Pages: 251 – 257

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n1.a15


Olgur Celikbas (Department of Mathematics,West Virginia University, Morgantown,W.V., U.S.A.)

Arash Sadeghi (School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran)

Yongwei Yao (Department of Mathematics and Statistics, Georgia State University, Atlanta, Ga., U.S.A.)


Let $R$ be a commutative Noetherian Cohen–Macaulay local ring that has positive dimension and prime characteristic. Li proved that the tensor product of a finitely generated non-free $R$-module $M$ with the Frobenius endomorphism ${}^{\varphi^n} \! R$ is not maximal Cohen–Macaulay provided that $M$ has rank and $n \gg 0$.We replace the rank hypothesis with the weaker assumption that $M$ is locally free on the minimal prime ideals of $R$. As a consequence, we obtain, if $R$ is a one-dimensional non-regular complete reduced local ring that has a perfect residue field and prime characteristic, then ${}^{\varphi^n} \! R \otimes_R {}^{\varphi^n} \! R $ has torsion for all $n \gg 0$. This property of the Frobenius endomorphism came as a surprise to us since, over such rings $R$, there exist non-free modules $M$ such that $M \otimes_R M$ is torsion-free.


Frobenius endomorphism, tensor product of modules, rank and torsion

2010 Mathematics Subject Classification

13A35, 13D07, 13H10

Received 28 June 2017

Received revised 17 October 2017

Published 21 February 2018